Quick revision notes · Mathematics 4024 · Chapter 27 of 35
Probability
Probability scale, single events, relative and expected frequency, sample spaces, tree diagrams and conditional probability.
At a glance
Probability scale, single events, relative and expected frequency, sample spaces, tree diagrams and conditional probability.
1Single events
- Probability is from 0 (impossible) to 1 (certain). Write it as a fraction, decimal or percentage.
- P(event) = number of ways it happens ÷ total number of equally likely outcomes.
- P(not A) = 1 − P(A). The probabilities of all possible outcomes add to 1.
Formulas
Worked example
The only outcomes A, B and C have probabilities 0.2, 0.45 and x. Find x.
- 0.2 + 0.45 + x = 1.
x = 0.35
2Relative and expected frequency
- Relative frequency = number of times it happened ÷ number of trials. It estimates the probability; more trials = better estimate.
- Expected frequency = probability × number of trials.
Worked example
A spinner landed on 6 in 45 of 300 spins. How many 6s would you expect in 1000 spins?
- Relative frequency = 45/300 = 0.15.
- 0.15 × 1000.
150
3Sample spaces and combined events
- A sample space diagram (grid) lists every outcome of two events — useful for two dice or two spinners.
- AND (independent events): multiply. OR (events that can't both happen): add.
- 'At least one' = 1 − P(none).
Formulas
Worked example
Two fair dice are thrown. Find the probability that the total is 7.
- 36 outcomes in the grid.
- Totals of 7: (1,6), (2,5), (3,4), (4,3), (5,2), (6,1) — 6 outcomes.
6/36 = 1/6
4Tree diagrams
- Each set of branches adds to 1.
- Multiply along a path; add the paths you want.
- With replacement: the second-stage probabilities are the same as the first.
- Without replacement: one fewer item in total, and one fewer of the colour already taken.
How to do it
- Draw the branches with probabilities.
- Multiply along each wanted path.
- Add the results of all wanted paths.
Worked example
A bag has 4 red and 6 blue sweets. Two are eaten without replacement. Find P(one of each colour).
- P(R then B) = 4/10 × 6/9 = 24/90.
- P(B then R) = 6/10 × 4/9 = 24/90.
- Add: 48/90.
8/15
5Conditional probability
- 'Given that' narrows the total to the group you are told about.
- From a table or Venn diagram: count only within the given group.
- From a tree: the second branches are already conditional on the first.
Formulas
Worked example
12 of 20 girls and 6 of 10 boys wear glasses. A student who wears glasses is chosen. Find the probability it is a boy.
- 18 students wear glasses.
- 6 of them are boys.
6/18 = 1/3
✗Common mistakes
| Don’t write | Write instead |
|---|---|
| Adding along the branches | Multiply along a path; add the results |
| Same denominator on both stages without replacement | 4/10 then 3/9 (or 6/9) |
| A probability bigger than 1 | Probabilities are between 0 and 1 |
| Writing a probability as '3 out of 5' or 3 : 5 | Write 3/5, 0.6 or 60% |
| Forgetting one of the two orders for 'one of each' | Red-then-blue AND blue-then-red |
★In the exam
- Draw the tree or grid even when not asked.
- Fractions don't need simplifying unless asked — but keep them exact.
- Check: all end probabilities of a tree add to 1.